What kind of stochastic model shall I build for x(t) if $stdev(dx)$ is linear to $dx$?

Question

I'm building a stochastic model for a process $x(t)$.

I notice that daily $dx$ and spot standard deviation of $dx$ seem not related. By spot standard deviation of $dx$ means i take 30 days $dx$ value and calculate the standard deviation.

However, if I average both by 1 year, they seem strongly linear related:

Any idea what kind of stochastic model shall i build? such as $dx = \mu dt + \sigma dW$, $dx / x = \mu dt + \sigma dt$, or $dx = (a-bx) dt + cx^\gamma dW$?